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On the stability of the motion of Saturn s Rings 288

Illustrations of the Dynamical Theory of Gases

On the Theory of Compound Colours and the Relations of the Colours of the
Spectrum

On the Theory of Three Primary Colours ***^

451

On Physical Lines of Force

On Reciprocal Figures and Diagrams of Forces °^*

A Dynamical Theory of the Electromagnetic Field 526

On the Calculation of the EquilibHum and Stiffness of Frames .... 598



ERRATA.



Page 40. In the first of equations (12), second group of terms, read
(hP dy' d^



instead of



d^^^d^^^d^^



with corresponding changes in the other two equations.

Page 153, five lines from bottom of page, read 127 instead of 276

Page 591, four lines from bottom of page the equation should be

d^M d2M_ldM
da? "^ db' a da~

Page 592, in the first line of the expression for L change

- K cos 26 into - ^ cosec 26.



[From the Proceedings of the Royal Society of Edinburgh, Vol, li. April, 1846.]



I. On the Description of Oval Curves, and those having a plurality of Foci; ivith
remarks by Professor Forbes. Communicated by Professor Forbes.



Mr Clerk Maxwell ingeniously suggests the extension of the common
theory of the foci of the conic sections to curves of a higher degree of com-
plication in the following manner : —

(1) As in the ellipse and hyperbola, any point in the curve has the
sum or difference of two lines drawn from two points or foci = a. constant
quantity, so the author infers, that curves to a certain degree analogous, may
be described and determined by the condition that the simple distance from
one focus pliLS a multiple distance from the other, may be = a constant quantity;
or more generally, m times the one distance + n times the other = constant.

(2) The author devised a simple mechanical means, by the wrapping
of a thread round pins, for producing these curves. See Figs. 1 and 2. He



Fig. 1. Two FocL Katios 1,



Fig. 2. Two Foci Ratios 2, 3.




then thought of extending the principle to other curves, whose property
should be, that the sum of the simple or multiple distances of any point of



DESCRIPTION OF OVAL CURVES.



the curve from three or more points or foci, should be = a constant quantity ;
and this, too, he has effected mechanically, by a very simple arrangement of
a string of given length passing round three or more fixed pins, and con-
straining a tracing point, P. See Fig. 3. Farther, the author regards curves

Fig. 3. Three Foci. Eatios of Equality.




of the first kind as constituting a particular class of curves of the second
kind, two or more foci coinciding in one, a focus in which two strings meet
being considered a double focus; when three strings meet a treble focus, &c.

Professor Forbes observed that the equation to curves of the first class is
easily found, having the form

V^+7= a-VhJ{x- c)' + y\

which is that of the curve known under the name of the First Oval of
Descartes*. Mr Maxwell had already observed that when one of the foci was
at an infinite distance (or the thread moved parallel to itself, and was confined
in respect of length by the edge of a board), a curve resembling an ellipse
was traced ; from which property Professor Forbes was led first to infer the
identity of the oval with the Cartesian oval, which is well known to have this
property. But the simplest analogy of all is that derived from the method of
description, r and r being the radients to any point of the curve from the two

foci ;

mr + nr — constant,

which in fact at once expresses on the undulatory theory of light the optical
character of the surface in question, namely, that light diverging from one
focus F without the medium, shall be correctly convergent at another point /

* Herschel, On Light, Art. 232 ; Lloyd, On Light and Vision, Chap. vii.



DESCRIPTION OF OVAL CURVES. J

within it ; and in this case the ratio — expresses the index of refraction of

the medium*.

If we denote by the power of either focus the number of strings leading
to it by Mr Maxwell's construction, and if one of the foci be removed to an
infinite distance, if the powers of the two foci be equal the curve is a parabola ;
if the power of the nearer focus be greater than the other, the curve is an
eUipse; if the power of the infinitely distant focus be the greater, the curve
is a hyperbola. The first case evidently corresponds to the case of the reflection
of parallel rays to a focus, the velocity being unchanged after reflection; the
second, to the refraction of parallel rays to a focus in a dense medium (in
which light moves slower) ; the third case to refraction into a rarer medium.

The ovals of Descartes were described in his Geometry, where he has also
given a mechanical method of describing one of themt, but only in a particular
case, and the method is less simple than Mr Maxwell's. The demonstration of
the optical properties was given by Newton in the Principia, Book i., prop. 97,
by the law of the sines; and by Huyghens in 1690, on the Theory of Undu-
lations in his Traite de la Lumiere. It probably has not been suspected that
so easy and elegant a method exists of describing these curves by the use of
a thread and pins whenever the powers of the foci are commensurable. For
instance, the curve. Fig. 2, drawn with powers 3 and 2 respectively, give the
proper form for a refracting surface of a glass, whose index of refraction is 1'50,
in order that rays diverging from f may be refracted to F.

As to the higher classes of curves with three or more focal points, we
cannot at present invest them with equally clear and curious physical properties,
but the method of drawing a curve by so simple a contrivance, which shall
satisfy the condition

mr + nr +pr" + &c. = constant,

is in itself not a little interesting; and if we regard, with Mr Maxwell, the
ovals above described, as the limiting case of the others by the coalescence
of two or more foci, we have a farther generalization of the same kind as that
so highly recommended by Montucla^ by which Descartes elucidated the conic
sections as particular cases of his oval curves.

♦ This was perfectly well shewn by Hnyghens in his Traite de la Lumiere, p. 111. (1690.)

+ Edit. 1683. Geometria, Lib. ii. p. 54.

X Histoire dea Mathematiqties. First Edit IL 102.



[From the Transactions of the Royal Society of Edinburgh, Vol. xvi. Part v.]



II. On the Theory of Rolling Curves. Communicated by the Eev. Professor

Kelland.



There is an important geometrical problem which proposes to find a curve
having a given relation to a series of curves described according to a given
law. This is the problem of Trajectories in its general form.

The series of curves is obtained from the general equation to a curve by
the variation of its parameters. In the general case, this variation may change
the form of the curve, but, in the case which we are about to consider, the
curve is changed only in position.

This change of position takes place partly by rotation, and partly by trans-
ference through space. The roUing of one curve on another is an example of
this compound motion.

As examples of the way in which the new curve may be related to the
series of curves, we may take the following : —

1. The new curve may cut the series of curves at a given angle. When
this angle becomes zero, the curve is the envelope of the series of curves.

2. It may pass through correspondiug points in the series of curves.
There are many other relations which may be imagined, but we shall confine
our attention to this, partly because it aSbrds the means of tracing various
curves, and partly on account of the connection which it has with many
geometrical problems.

Therefore the subject of this paper will be the consideration of the relations
of three curves, one of which is fixed, while the second rolls upon it and
traces the third. The subject of rolling curves is by no means a new one.
The first idea of the cycloid is attributed to Aristotle, and involutes and
evolutes have been long known.



THE THEORY OF ROLLING CURVES.

In the Histmy of the Royal Academy of Sciences for 1704, page 97,
there is a memoir entitled "Nouvelle formation des Spirales," by M. Varignon,
in which he shews how to construct a polar curve from a curve referred to
rectangular co-ordinates by substituting the radius vector for the abscissa, and
a circular arc for the ordinate. After each curve, he gives the curve into
which it is " unrolled," by which he means the curve which the spiral must
be rolled upon in order that its pole may trace a straight line; but as this
18 not the principal subject of his paper, he does not discuss it very fully.

There is also a memoir by M. de la Hire, in the volume for 1706, Part ii.,
page 489, entitled "Methode generale pour r^duire toutes les Lignes courbes ^
des Roulettes, leur generatrice ou leur base ^tant donnde telle qu'on voudra."

M. de la Hire treats curves as if they were polygons, and gives geome-
trical constructions for finding the fixed curve or the rolling curve, the other
two being given; but he does not work any examples.

In the volume for 1707, page 79, there is a paper entitled, "Methode
generale pour determiner la nature des Courbes form^es par le roulement de
toutes sortes de Courbes sur une autre Courbe quelconque." Par M. Nicole.

M. Nicole takes the equations of the three curves referred to rectangular
co-ordinates, and finds three general equations to connect them. He takes the
tracing-point either at the origin of the co-ordinates of the rolled curve or not.
He then shews how these equations may be simplified in several particular
cases. These cases are —

(1) When the tracing-point is the origin of the roUed curve.

(2) When the fixed curve is the same as the rolling cxirve.

(3) When both of these conditions are satisfied.

(4) When the fixed line is straight.

He then says, that if we roll a geometric curve on itself, we obtain a new
geometric curve, and that we may thus obtain an infinite number of geometric
curves.

The examples which he gives of the application of his method are all taken
from the cycloid and epicycloid, except one which relates to a parabola, rolling
on itself, and tracing a cissoid with its vertex. The reason of so small a
number of examples being worked may be, that it is not easy to eliminate
the co-ordinates of the fixed and rolling curves from his equations.

The case in which one curve roUing on another produces a circle is treated
of in Willis's Principles of Mechanism. Class C. Boiling Contact.



6 THE THEORY OP ROLLHiTO CURVES.

He employs the same method of finding the one curve from the other
which is used here, and he attributes it to Euler (see the Acta Petropolitana,
Vol. v.).

Thus, nearly all the simple cases have been treated of by different authors;
but the subject is still far from being exhausted, for the equations have been
applied to very few curves, and we may easily obtain new and elegant proper-
ties from any curve we please.

Almost all the more notable curves may be thus linked together in a great
variety of ways, so that there are scarcely two curves, however dissimilar,
between which we cannot form a chain of connected curves.

This will appear in the list of examples given at the end of this paper.




Let there be a curve KAS, whose pole is at C.



THE THEORY OF ROLLING CURVES. 7

Let the angle DCA = 6, and CA=r, and let

Let this curve remain fixed to the paper.

Let there be another curve BAT, whose pole is B.

Let the angle MBA = 0t, and BA=r^, and let

Let this curve roll along the curve KAS without slipping.
Then the pole B will describe a third curve, whose pole is C.
Let the angle DCB = 0^, and CB = r„ and let

We have here six unknown quantities 0,dAr,r^r^; but we have only three
equations given to connect them, therefore the other three must be sought for
in the enunciation.

But before proceeding to the investigation of these three equations, we must
premise that the three curves will be denominated as follows : —
The Fixed Curve, Equation, e^ = ^^{r^.
The Rolled Curve, Equation, 0. = <f>,{r,).
Tlie Traced Curve, Equation, 6^ = 4>.,{r^.

When it is more convenient to make use of equations between rectangular
co-ordinates, we shall use the letters x^^, x^^, x^ij^. We shall always employ the
letters s^s^^ to denote the length of the curve from the pole, p.p^p^ for the per-
pendiculars from the pole on the tangent, and q^q/i^ for the intercepted part of
the tangent.

Between these quantities, we have the following equations: —

r = ^/^T?, ^ = tan-|,

a? = r cos ^, y = r sin 6,

r" ydx — xdy



jm'S ""^w+w'



THE THEORY OF ROLLING CURVES.

rdr

dS _ xdx + ydy



2=-r=7x!fi' r-



J{dxy + (dyY'



' "^ W '^d^ daf

We come now to consider the three equations of rolling which are involved
in the enunciation. Since the second curve rolls upon the first without slipping,
the length of the fixed curve at the point of contact is the measure of the
length of the rolled curve, therefore we have the following equation to connect
the fixed curve and the rolled curve —

«! = Sj.

Now, by combining this equation with the two equations

it is evident that from any of the four quantities 6{r^6^r^ or x^^x^^, we can
obtain the other three, therefore we may consider these quantities as known
functions of each other.

Since the curve rolls on the fixed curve, they must have a common tangent.

Let PA be this tangent, draw BP, CQ perpendicular to PA, produce CQ,
and draw BR perpendicular to it, then we have CA=r^, BA = r^, and CB = r,;
CQ=p„ PB=p,, and BN=p,; AQ = q„ AP = q„ and CN=q,.

Also r,'=CR=CR + RR = (CQ + PBY+(AP-AQf

=p,' + 2p,p, +p,' + r,' -p,' - 2q,q, + r," -p,'
fz = n' + n' + 2piPa - 2q,q^.
Since the first curve is fixed to the paper, we may find the angle 6,.
Thus e, = DCB = DCA + ACQ + RCB

= e?. + tan-| + tan-|§
^, = ^, + tan - ^ + tan-^ ^^^^



TjdO^ Pi +pi



THE THEORY OF ROLLING CURVES. »

Thus we have found three independent equations, which, together with the
equations of the curves, make up six equations, of which each may be deduced
from the others. There is an equation connecting the radii of curvature of the
three curves which is sometimes of use.

The angle through which the rolled curve revolves during the description of
the element ds„ is equal to the angle of contact of the fixed curve and the
rolling curve, or to the sum of their curvatures,

ds^ ds^ ds.

But the radius of the rolled curve has revolved In the opposite direction
through an angle equal to dO,, therefore the angle between two successive posi-
tions of r, is equal to -^-dd,. Now this angle is the angle between two

successive positions of the normal to the traced curve, therefore, if be the
centre of curvature of the traced curve, it is the angle which ds^ or ds^ subtends
at 0. Let OA^T, then

ds^ r4d^ ds, ,^ _ ds^ ds, ,.

^J__J_ 1 _^
•*• '^'ds, T~ R, R, ds/

-tAt^tJ RJR.'

As an example of the use of this equation, we may examine a property
of the logarithmic spiral.

In this curve, p = mr, and R = — , therefore if the rolled curve be the
■^ m

logarithmic spiral

/I 1\ 1 ^m

"^[t^tJ-r^v/

m_ 1

t~r:,*

AO
therefore ^0 in the figure = ?ni2i, and -^ = m.

Let the locus of 0, or the evolute of the traced curve LYBH, be the
curve OZY, and let the evolute of the fixed curve KZAS be FEZ, and let
us consider FEZ as the fixed curve, and OZF as the traced curve.



10 THE THEORY OF ROLLING CURVES.

Then in the triangles BPA, AOF, we have OAF=PBA, and ^='^ = ^y

therefore the triangles are similar, and FOA = APB = - , therefore OF is perpen-
dicular to OA, the tangent to the curve OZY, therefore OF is the radius of
the curve which when roUed on FEZ traces OZY, and the angle which the
curve makes with this radius is OFA=PAB = %mr^m, which is constant, there-
fore the curve, which, when rolled on FEZ, traces OZY, is the logarithmic
spiral. Thus we have proved the following proposition : " The involute of the
curve traced by the pole of a logarithmic spiral which rolls upon any curve,
is the curve traced by the pole of the same logarithmic spiral when rolled on
the involute of the primary curve."

It follows from this, that if we roll on any curve a curve having the
property _2:»i — Wjri, and roll another curve having Pi = 'm^r^ on the curve traced,
and so on, it is immaterial in what order we roll these curves. Thus, if we
roll a logarithmic spiral, in which jp = mr, on the nth involute of a circle whose
radius is a, the curve traced is the w+lth involute of a circle whose radius
is Jl-m\

Or, if we roll successively m logarithmic spirals, the resulting curve is the
n + mth involute of a circle, whose radius is

aJl—m^ sll- m/, Jkc.

We now proceed to the cases in which the solution of the problem may
be simplified. This simplification is generally effected by the consideration that
the radius vector of the rolled curve is the normal drawn from the traced
curve to the fixed curve.

In the case in which the curve is rolled on a straight line, the perpen-
dicular on the tangent of the rolled curve is the distance of the tracing point
from the straight line ; therefore, if the traced curve be defined by an equation
in iCg and y„

'^.°p.= / "'„... (1)'



and



'••=^'^©^ ^'^-



THE THEORY OF ROLLING CURVES. 11

By substituting for r, in the first equation, its value, as derived from the
second, we obtain

-■©■[©■-]=©'■

If we know the equation to the rolled curve, we may find (-7-^') in

terms of r,, then by substituting for r, its value in the second equation, we

dx (1 X

have an equation containing x^ and -^, from which we find the value of -t— '

dy, du,

in terms of x^; the integration of this gives the equation of the traced curve.

As an example, we may find the curve traced by the pole of a hyperbolic
spiral which rolls on a straight line.

a

fdrA' _ rl
,ddj ~ a'



The equation of the rolled curve is 6^ =



- •■©■-■[(IJ-]'



dx^ _ ^3
'* dy,~Ja'-x,''

This is the differential equation of the tractory of the straight line, which
is the curve traced by the pole of the hyperbolic spiral.
By eliminating x^ in the two equations, we obtain

dr^_ /dxA

This equation serves to determine the rolled curve when the traced cuive
is given.

As an example we shall find the curve, which being rolled on a straight
line, traces a common catenary.

Let the equation to the catenary be



'l(e' + e-^.



12 THE THEORY OF ROLLING CURVES.



Then



dy,~N a' '



dr



then by integration ^ =cos'^ ( 1 j



2a
r =



1+COS0'



This is the polar equation of the parabola, the focus being the pole ; there-
fore, if we roll a parabola on a straight line, its focus will trace a catenary.

The rectangiilar equation of this parabola is af = Aay, and we shall now
consider what curve must be rolled along the axis of y to trace the parabola.

By the second equation (2),

n = ^9 /-4- + l> but x^^Pi,
V ^»

.-. r/=^/ + 4a",
.-. 2a = Vr/-jp/ = g'„
but q^ is the perpendicular on the normal, therefore the normal to the curve
always touches a circle whose radius is 2a, therefore the curve is the involute
of this circle.

Therefore we have the following method of describing a catenary by con-
tinued motion.

Describe a circle whose radius is twice the parameter of the catenary; roll a
straight line on this circle, then any point in the line will describe an involute



THE THEORY OF ROLLING CURVES. 13

of the circle ; roll this curve on a straight line, and the centre of the circle will
describe a parabola ; roll this parabola on a straight line, and its focus will trace
the catenary required.

We come now to the case in which a straight line rolls on a curve.

When the tracing-point is in the straight line, the problem becomes that
of involutes and evolutes, which we need not enter upon ; and when the tracmg-
point is not in the straight line, the calculation is somewhat complex; we shall
therefore consider only the relations between the curves described in the first
and second cases.

Definition. — The curve which cuts at a given angle all the circles of a
given radius whose centres are in a given curve, is called a tractory of the
given curve.

Let a straight line roll on a curve A, and let a point in the straight
line describe a curve B, and let another point, whose distance from the first
point is b, and from the straight line a, describe a curve C, then it is evident
that the curve B cuts the circle whose centre is in C, and whose radius is b,

at an angle whose sine is equal to r, therefore the curve 5 is a tractory of

the curve C.

When a = b, the curve B is the orthogonal tractory of the curve C. If
tangents equal to a be drawn to the curve B, they will be terminated in
the curve C; and if one end of a thread be carried along the curve C, the
other end will trace the curve B.

When a = 0, the curves B and C are both involutes of the curve A,
they are always equidistant from each other, and if a circle, whose radius is
6, be rolled on the one, its centre will trace the other.

If the curve A is such that, if the distance between two points measured
along the curve is equal to 6, the two points are similarly situate, then the
curve B is the same with the curve C. Thus, the curve A may be a re-
entrant curve, the circumference of which is equal to 6.

When the curve -4 is a circle, the curves B and C are always the same.

The equations between the radii of curvature become

1 1 _ r



14 THE THEORY OF ROLLING CURVES.

When a = 0, T=0, or the centre of curvature of the curve B is at the
point of contact. Now, the normal to the curve C passes through this point,
therefore —

"The normal to any curve passes through the centre of curvature of its
tractory,"

In the next case, one curve, by rolling on another, produces a straight
line. Let this straight line be the axis of y, then, since the radius of the
rolled curve is perpendicular to it, and terminates in the fixed curve, and
since these curves have a common tangent, we have this equation,

If the equation of the rolled curve be given, find -j-^ in terms of r^, sub-
stitute Xi for r^, and multiply by x^, equate the result to -^ , and integrate.

Thus, if the equation of the rolled curve be

d = Ar-"" + &c. + Kr-^ + Lr'^ + if log r + iVr + &c. + Zr"",

^ = - n^r-(»+^) - &c. - 2Kr-' - I/p-' + Mr'' + N+ &c. + wZr"-^
dr

-r-= - nAx~'* - &c. - 2Kx~"- - Lx~^ + M+ Nx + &c. + nZx",
ax

y = -^ Aa^-"" + &c. + 2Kx-' -L\ogx + Mx + ^Naf + &c. + -^ Zx""^',

which is the equation of the fixed curve.

If the equation of the fixed curve be given, find -^ in terms of cc, sub-
stitute r for X, and divide by r, equate the result to -t-, and integrate.

Thus, if the fixed curve be the orthogonal tractory of the straight line,
whose equation is



y = a log . + Ja^

a + \la^ — x^

dy _ Jo' — af
dx~ X



THE THEORY OF ROLUNG CURVES.



15



de _ Ja?-7*
dr r*

= cos"^






this is the equation to the orthogonal tractory of a circle whose diameter is
equal to the constant tangent of the fixed curve, and its constant tangent
equal to half that of the fixed curve.

This property of the tractory of the circle may be proved geometrically,
thus — Let P be the centre of a circle whose radius is PD, and let CD be
a line constantly equal to the radius. Let BCP be the curve described by
the point C when the point D is moved along the circumference of the circle,
then if tangents equal to CD be drawn to the curve, their extremities will
be in the circle. Let ACH be the curve on which BCP rolls, and let OPE
be the straight line traced by the pole, let CDE be the common tangent,
let it cut the circle in D, and the straight line in E.




Then CD = PD, .'. LDCP^ LDPC, and CP is perpendicular to OE,
.'. L CPE= LDCP+ LDEP. Take away LDCP-^ L DPC, and there remains
DPE=DEP, .-. PD=^DE, .-. CE=2PD.



16 THE THEORY OF ROLLING CURVES.

Therefore the curve ACH haa a constant tangent equal to the diameter of
the circle, therefore ACH is the orthogonal tractorj of the straight line, which
is the tractrix or equitangential curve.

The operation of finding the fixed curve from the rolled curve is what
Sir John Leslie calls " divesting a curve of its radiated structure."

The method of finding the curve which must be rolled on a circle to
trace a given curve is mentioned here because it generally leads to a double
result, for the normal to the traced curve cuts the circle in two points, either
of which may be a point in the rolled curve.



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